Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 38C / a / Solution

Balancing the downslope gravitational force density $\rho g\sin \alpha$ against the viscous term $\mu u_{yy}$ gives the velocity scaleIf the streamwise variation scale is $L\sim h/|h_x|$, the lubrication theory requirement is firstThe ratio of streamwise inertia $\rho U^2/L$ to transverse viscous stress $\mu U/h^2$ isThus self-consistency also requires